A fit with BAT.jl

In this example we show how to bring the PDF parametrisation and forward model together with BAT.jl to perform a fit of simulated data. This fit is a work in progress and just a starting point for verification of the method.

using BAT, DensityInterface
using PartonDensity
using QCDNUM
using Plots, Random, Distributions, ValueShapes, ParallelProcessingTools

Simulate some data

seed = 42
Random.seed!(seed) # for reproducibility

MersenneTwister(42)

We can start off by simulating some fake data for us to fit. This way, we know exactly what initial conditions we have specified and can check the validity of our inference, assuming the generative model is the one that is producing our data.

This is a good first check to work with.

Specify the input PDFs

See the Input PDF parametrisation and priors example for more information on the definition of the input PDFs.

pdf_params = PDFParameters(λ_u=0.7, K_u=4.0, λ_d=0.5, K_d=6.0,
    λ_g1=0.7, λ_g2=-0.4, K_g=6.0, λ_q=-0.5, weights=[1, 0.5, 0.3, 0.2, 0.1, 0.1, 0.1]);

#plot_input_pdfs(pdf_params)

PDFParameters
  λ_u: Float64 0.7
  K_u: Float64 4.0
  λ_d: Float64 0.5
  K_d: Float64 6.0
  λ_g1: Float64 0.7
  λ_g2: Float64 -0.4
  K_g: Float64 6.0
  λ_q: Float64 -0.5
  seed: Int64 0
  weights: Array{Float64}((7,)) [1.0, 0.5, 0.3, 0.2, 0.1, 0.1, 0.1]
  θ: Array{Float64}((7,)) [0.21430506549299336, 0.28298257876412375, 0.011738407805513947, 0.17869065609183865, 6.810183645202651e-7, 1.9090574971298878e-6, 1.528257884562461e-11]

Go from PDFs to counts in ZEUS detector bins

Given the input PDFs, we can then evolve, calculate the cross sections, and fold through the ZEUS transfer matrix to get counts in bins. Here, we make use of some simple helper functions to do so. For more details, see the Forward model example.

first specify QCDNUM inputs

qcdnum_grid = QCDNUMGrid(x_min=[1.0e-3], x_weights=[1], nx=100,
    qq_bounds=[1.0e2, 3.0e4], qq_weights=[1.0, 1.0], nq=50, spline_interp=3)
qcdnum_params = QCDNUMParameters(order=2, α_S=0.118, q0=100.0, grid=qcdnum_grid,
    n_fixed_flav=5, iqc=1, iqb=1, iqt=1, weight_type=1);

now SPLINT and quark coefficients

splint_params = SPLINTParameters();
quark_coeffs = QuarkCoefficients();

initialise QCDNUM

forward_model_init(qcdnum_grid, qcdnum_params, splint_params)

  +---------------------------------------------------------------------+
  |                                                                     |
  |    If you use QCDNUM, please refer to:                              |
  |                                                                     |
  |    M. Botje, Comput. Phys. Commun. 182(2011)490, arXiV:1005.1481    |
  |                                                                     |
  +---------------------------------------------------------------------+



 FILLWT: start unpolarised weight calculations
 Subgrids    1 Subgrid points  100
 Pij LO
 Pij NLO
 Pij NNLO
 Aij LO
 Aij NLO
 Aij NNLO
 FILLWT: weight calculations completed


 ZMFILLW: start weight calculations   4  38   0   0
 ZMFILLW: calculations completed

  +---------------------------------------+
  | You are using SPLINT version 20210919 |
  +---------------------------------------+

run forward model

counts_pred_ep, counts_pred_em = forward_model(pdf_params, qcdnum_params,
    splint_params, quark_coeffs);

take a poisson sample

nbins = size(counts_pred_ep)[1]
counts_obs_ep = zeros(UInt64, nbins)
counts_obs_em = zeros(UInt64, nbins)

for i in 1:nbins
    counts_obs_ep[i] = rand(Poisson(counts_pred_ep[i]))
    counts_obs_em[i] = rand(Poisson(counts_pred_em[i]))
end

plot(1:nbins, counts_pred_ep, label="Expected counts (eP)", color="blue")
plot!(1:nbins, counts_pred_em, label="Expected counts (eM)", color="red")
scatter!(1:nbins, counts_obs_ep, label="Detected counts (eP)", color="blue")
scatter!(1:nbins, counts_obs_em, label="Detected counts (eM)", color="red")
plot!(xlabel="Bin number")

store

sim_data = Dict{String, Any}()
sim_data["nbins"] = nbins;
sim_data["counts_obs_ep"] = counts_obs_ep;
sim_data["counts_obs_em"] = counts_obs_em;

write to file

pd_write_sim("output/simulation.h5", pdf_params, sim_data)

Fit the simulated data

Now we can try to fit this simulated data using Bat.jl. The first step is to define the prior and likelihood. For now, let's try relatively narrow priors centred on the true values.

prior = NamedTupleDist(
    θ = Dirichlet([1, 0.5, 0.3, 0.2, 0.1, 0.1, 0.1]),
    λ_u = Truncated(Normal(pdf_params.λ_u, 0.2), 0, 1),
    K_u = Truncated(Normal(pdf_params.K_u, 1), 2, 10),
    λ_d = Truncated(Normal(pdf_params.λ_d, 0.2), 0, 1),
    K_d = Truncated(Normal(pdf_params.K_d, 1), 2, 10),
    λ_g1 = Truncated(Normal(pdf_params.λ_g1, 0.2), 0, 1),
    λ_g2 = Truncated(Normal(pdf_params.λ_g2, 0.2), -1, 0),
    K_g =  Truncated(Normal(pdf_params.K_g, 1), 2, 10),
    λ_q = Truncated(Normal(pdf_params.λ_q, 0.1), -1, 0),
);

The likelihood is similar to that used in the input PDF parametrisation example. We start by accessing the current parameter set of the sampler's iteration, then running the forward model to get the predicted counts and comparing to the observed counts using a simple Poisson likelihood.

The @critical macro is used because forward_model() is currently not thread safe, so this protects it from being run in parallel.

likelihood = let d = sim_data

    counts_obs_ep = d["counts_obs_ep"]
    counts_obs_em = d["counts_obs_em"]
    nbins = d["nbins"]

    logfuncdensity(function (params)

            pdf_params = PDFParameters(λ_u=params.λ_u, K_u=params.K_u, λ_d=params.λ_d,
                                       K_d=params.K_d, λ_g1=params.λ_g1, λ_g2=params.λ_g2,
                                       K_g=params.K_g, λ_q=params.λ_q, θ=Vector(params.θ))

            @critical counts_pred_ep, counts_pred_em = forward_model(pdf_params,
                qcdnum_params, splint_params, quark_coeffs);

            ll_value = 0.0
            for i in 1:nbins
                ll_value += logpdf(Poisson(counts_pred_ep[i]), counts_obs_ep[i])
                ll_value += logpdf(Poisson(counts_pred_em[i]), counts_obs_em[i])
            end

            return ll_value
    end)
end

LogFuncDensity(Main.var"#1#2"{Int64, Vector{UInt64}, Vector{UInt64}}(153, UInt64[0x000000000000022f, 0x00000000000001cd, 0x0000000000000179, 0x000000000000015d, 0x00000000000000f7, 0x000000000000010b, 0x00000000000000d8, 0x00000000000001f4, 0x000000000000023f, 0x0000000000000208  …  0x000000000000001e, 0x0000000000000016, 0x0000000000000012, 0x000000000000000b, 0x0000000000000007, 0x0000000000000003, 0x0000000000000003, 0x0000000000000000, 0x000000000000006b, 0x0000000000000008], UInt64[0x0000000000000176, 0x0000000000000153, 0x0000000000000117, 0x000000000000010d, 0x00000000000000c6, 0x00000000000000c0, 0x0000000000000099, 0x000000000000019b, 0x00000000000001ae, 0x000000000000017a  …  0x0000000000000015, 0x0000000000000012, 0x0000000000000009, 0x0000000000000008, 0x0000000000000002, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x000000000000002f, 0x0000000000000003]))

We can now run the MCMC sampler. We will start by using the Metropolis-Hastings algorithm as implemented in BAT.jl. To demonstrate, we run a short chain of 10^3 iterations, which should take around 10 minutes.

posterior = PosteriorDensity(likelihood, prior);
samples = bat_sample(
    posterior,
    MCMCSampling(mcalg=MetropolisHastings(), nsteps=10^3, nchains=2)
).result;

[ Info: Initializing new RNG of type Random123.Philox4x{UInt64, 10}
[ Info: Using transform algorithm BAT.PriorSubstitution()
[ Info: Trying to generate 2 viable MCMC chain(s).
[ Info: Selected 2 MCMC chain(s).
[ Info: Begin tuning of 2 MCMC chain(s).
[ Info: MCMC Tuning cycle 1 finished, 2 chains, 2 tuned, 2 converged.
[ Info: MCMC tuning of 2 chains successful after 1 cycle(s).
[ Info: Running post-tuning stabilization steps for 2 MCMC chain(s).

Let's save the result for further analysis

import HDF5
bat_write("output/results.h5", samples)

Analysing the results

First, let's load our simulation inputs and results

pdf_params, sim_data = pd_read_sim("output/simulation.h5");
samples = bat_read("output/results.h5").result;

We can check some diagnostics using built in BAT.jl, such as the effective sample size shown below

bat_eff_sample_size(unshaped.(samples))[1]

15-element Vector{Float64}:
  12.955447679597027
  29.855010597487173
  44.767155812146456
  38.06414069360615
  27.44411306616384
  66.16195369992882
  21.025026019758513
 101.02302170293379
  59.92456184579444
  67.62578115078692
  30.44530770883487
  38.03609581987014
  33.53018062648647
  24.535647518294148
  34.56492997960154

We see a value for each of our 15 total parameters. As the Metropolis-Hastings algorithm's default implementation isn't very efficient, we see that the effective sample size is only a small percentage of the input nsteps. We should try to improve this if possible, or use a much larger nsteps value.

For demonstration purposes, we will continue to show how we can visualise the results in this case. For robust inference, we need to improve the sampling stage above.

We can use BAT.jl's built in plotting recipes to show the marginals, for example, consider λ_u, and compare to the known truth.

plot(
    samples, :(λ_u),
    nbins=50,
    colors=[:skyblue4, :skyblue3, :skyblue1],
    alpha=0.7,
    marginalmode=false,
    legend=:topleft
)
vline!([pdf_params.λ_u], color="black", label="truth", lw=3)

Rather than making a large plot 15 different marginals, it can be more useful to visualise the posterior distribution in differently, such as the shape of the distributions we are trying to fit, or the model space. Helper functions exist for doing just this.

Using BAT recipe

function wrap_xtotx(p::NamedTuple{(:K_d, :K_g, :K_u, :θ, :λ_d, :λ_g1,
                                   :λ_g2, :λ_q, :λ_u)}, x::Real)
    pdf_params = PDFParameters(λ_u=p.λ_u, K_u=p.K_u, λ_d=p.λ_d, K_d=p.K_d, λ_g1=p.λ_g1,
        λ_g2=p.λ_g2, K_g=p.K_g, λ_q=p.λ_q, θ=p.θ)
    return xtotx(x, pdf_params)
end

x_grid = range(1e-3, stop=1, length=50)
plot(x_grid, wrap_xtotx, samples, colors=[:skyblue4, :skyblue3, :skyblue1],
     legend=:topright)
plot!(x_grid, [xtotx(x, pdf_params) for x in x_grid], color="black", lw=3,
      label="Truth", linestyle=:dash)
plot!(xaxis=:log)

Using PartonDensity.jl

plot_model_space(pdf_params, samples)

Alternatively, we can also visualise the implications of the fit in the data space, as shown below.

plot_data_space(pdf_params, sim_data, samples, qcdnum_grid, qcdnum_params,
                splint_params, quark_coeffs)